Rudi Skacel struck a dramatic late winner as Hearts beat St Mirren in the SPL while there were also victories for Dundee United and Kilmarnock.
The Czech slammed home a Craig Thomson corner in injury time just minutes after Ryan Stevenson had equalised.
Higdon gave the visitors a deserved half-time lead with a 15th-minute header to make up for an earlier miss but Skacel's dipping effort brought Hearts level in the 55th minute.
Stevenson then cancelled out Higdon's second before Skacel pounced in the box to seal Hearts' first win in four games.
Dundee United guaranteed a top-six place with a 1-0 victory over Inverness Caledonian Thistle.
The only goal of the match came two minutes before the break when Danny Swanson guided the ball home with a smart left-foot finish.
It was United's fifth game in 12 days but they showed no signs of tiredness against the Highlanders as they won five consecutive league games for the first time in 14 years.
Kilmarnock took a huge step towards securing a top-six place as they came from behind to beat Motherwell 3-1 with the help of a controversial penalty.
John Sutton slotted Motherwell into an early lead but Manuel Pascali headed an equaliser before winning a soft penalty when he fell under an aerial challenge from Gavin Gunning.
Jamie Hamill converted the 43rd-minute spot-kick and Killie dominated the second half after Liam Kelly headed home at the near post six minutes after the break.
The victory moved Kilmarnock eight points above Motherwell with just three games left until the SPL split.
Hamilton's home hoodoo continued as they failed to break the deadlock against St Johnstone, though they did move a point closer to safety.
Perth defender Danny Grainger struck the outside of the post with a free-kick in the first half before Saints keeper Peter Enckelman made a couple of fine saves from Ali Crawford and Flavio Paixao.
In the final moments, after the visitors had survived a couple of goal attempts, Accies defender Simon Mensing was shown a straight red card for a challenge on Chris Millar.
But although the match ended with the SPL's bottom side still looking for their first home league win of the season, they moved to within six points of second-bottom St Mirren, although the Paisley side have a game in hand.
